Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include<bits/stdc++.h>
#define ll long long int
#define fu(a,b) for(int i=a;i<b;i++)
#define fv(a,b) for(int j=a;j<b;j++)

using namespace std;
double dp[4000];
int main()
{
	int n;
	cin>>n;
	double p;
	
	dp[0]=1;
	fu(0,n)
	{
		cin>>p;
		for(int j=i+1;j>=0;j--)
		{
			dp[j]=(j==0?0:dp[j-1]*p)+dp[j]*(1-p);
		}
	}
	double ans=0;
	fu(n/2+1,n+1){
		ans+=dp[i];
	}
	cout<<setprecision(10)<<fixed<<ans;
}